Weakly separative weakly commutative semigroups |
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Authors: | Attila Nagy |
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Institution: | 1. Department of Algebra, Institute of Mathematics, Budapest University of Technology and Economics, Pf. 91, 1521, Budapest, Hungary
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Abstract: | A semigroup S is called a weakly commutative semigroup if, for every a,b∈S, there is a positive integer n such that (ab) n ∈Sa∩bS. A semigroup S is called archimedean if, for every a,b∈S, there are positive integers m and n such that a n ∈SbS and b m ∈SaS. It is known that every weakly commutative semigroup is a semilattice of weakly commutative archimedean semigroups. A semigroup S is called a weakly separative semigroup if, for every a,b∈S, the assumption a 2=ab=b 2 implies a=b. In this paper we show that a weakly commutative semigroup is weakly separative if and only if its archimedean components are weakly cancellative. This result is a generalization of Theorem 4.16 of Clifford and Preston (The Algebraic Theory of Semigroups, Am. Math. Soc., Providence, 1961). |
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